A053725 -id:A053725 - cp's OEIS frontend

A053722 Number of n X n binary matrices of order dividing 2 (also number of solutions to X^2=I in GL(n,2)).

1, 4, 22, 316, 6976, 373024, 32252032, 6619979776, 2253838544896, 1810098020122624, 2442718932612677632, 7758088894129169760256, 41674675294431186817908736, 526370120583359572695165435904, 11281778621698661853306239290703872

Offset: 1

Vladeta Jovovic, Mar 23 2000

nonn

In characteristic 2, A^2 = I if and only if B^2 = 0 where B = I + A, so a(n) is also equal to the number of n X n binary matrices B such that B^2 = 0.

Conjecture: the two matrices I and 0 have the largest number of square roots. Checked for n=1..5. - Alexey Slizkov, Jan 11 2024

Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Robert Israel, Table of n, a(n) for n = 1..81
Jason Fulman and C. Ryan Vinroot, Generating functions for real character degree sums of finite general linear and unitary groups, arXiv:1306.0031 [math.GR], 2013 (see Theorem 3.4 for a g.f.).
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A053725, A063505, A225371.

Q:= Product(1+u/2^i,i=1..infinity)/Product(1-u^2/2^i,i=1..infinity):
S:= series(Q,u,31):
seq(coeff(S,u,n)*mul(2^i-1,i=1..n), n=1..30); # Robert Israel, Mar 26 2018

QP = QPochhammer; Q = (1-x) QP[-x, 1/2]/QP[x^2, 1/2];
Table[(-1)^n QP[2, 2, n] SeriesCoefficient[Q, {x, 0, n}], {n, 1, 14}] (* Jean-François Alcover, Sep 17 2018, from Maple *)

g = lambda n: GL(n,2).order() if n>0 else 1
a053722 = lambda n: g(n)*sum(1/(g(k)*g(n-2*k)*2**(k**2+2*k*(n-2*k))) for k in range(1+floor(n/2))) if n>0 else 0
map(a053722, range(25))
# Dmitrii Pasechnik, Oct 02 2015

a(n) = Sum_{k=0..floor(n/2)} (2^n - 1)(2^{n-1} - 1) ... (2^{n-2k+1}-1) * 2^{k(k-1)/2} / ((2^k - 1)(2^{k-1} - 1) ... (2^1 - 1)). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 05 2001 [Corrected using the paper by Morrison, which also mentions that there is an error in this entry. k = 0 contributes 1 to the sum. If omitted, this gives the number of matrices of order exactly 2. Jan Kristian Haugland, Apr 24 2024]

A063393 Number of solutions of x^10=1 in general affine group AGL(n,2).

Original entry on oeis.org

2, 10, 92, 23200, 21391520, 35841831040, 95709758320640, 6206883395497062400, 1502803598296957497344000, 654083813715060854940290252800, 450433384822340709737677746549555200

Offset: 1

Vladeta Jovovic, Jul 16 2001

nonn

Cf. A063385-A063393, A062710, A053722, A053725, A053718, A053770-A053777.

A063385 Number of solutions of x^2=1 in general affine group AGL(n,2).

Original entry on oeis.org

2, 10, 92, 1696, 59552, 4124800, 556101632, 148425895936, 78099471368192, 81705857229783040, 169694608681978560512, 702657511446831375056896, 5797142351555426979908943872, 95500953266115919784543392890880, 3140561514292519005433439594146168832

Offset: 1

Vladeta Jovovic, Jul 16 2001

nonn

Cf. A063386-A063393, A062710, A053722, A053725, A053718, A053770-A053777.

More terms from Sean A. Irvine, Apr 23 2023

A053770 Number of n X n binary matrices of order dividing 5 (i.e., number of solutions of X^5=I in GL(n,2)).

Original entry on oeis.org

1, 1, 1, 1345, 666625, 223985665, 65019838465, 105072058957825, 11436238073940148225, 997931868985434228916225, 74706800043914446529756135425, 5321514758546715999509008953114625, 3721818216683598164434468712927276826625

Offset: 1

Vladeta Jovovic, Mar 24 2000

nonn

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A053722, A053725, A053718.

\\ See A053725 for F(n,q,k).
F(15, 2, 5) \\ Andrew Howroyd, Jul 09 2018

a(13) from Andrew Howroyd, Jul 09 2018

A053777 Number of n X n binary matrices of order dividing 12 (i.e., number of solutions of X^12=I in GL(n,2)).

Original entry on oeis.org

1, 6, 120, 10368, 2582208, 3143720448, 11692182896640, 219197554267521024, 12804488375721592356864, 3325324798296500862330077184, 2537067900325971750395878897090560, 8900626797123384385697033838119859781632, 65799342288255766009804607851267459830106816512

Offset: 1

Vladeta Jovovic, Mar 24 2000

nonn

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A053722, A053725, A053718.

More terms from Sean A. Irvine, Jan 16 2022

A053771 Number of n X n binary matrices of order dividing 6 (i.e., number of solutions of X^6=I in GL(n,2)).

Original entry on oeis.org

1, 6, 78, 6588, 1332288, 1335398688, 2230748717184, 13819713971871744, 219439188546028498944, 16360198814356838801178624, 3333281205541847127897252298752, 2704161270841324410691567986117967872

Offset: 1

Vladeta Jovovic, Mar 24 2000

nonn

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A053722, A053725, A053718.

A053773 Number of n X n binary matrices of order dividing 8 (i.e., number of solutions of X^8=I in GL(n,2)).

Original entry on oeis.org

1, 1, 4, 64, 4096, 1048576, 1073741824, 4398046511104, 72057594037927936, 1989505896802466922496, 164384949539438492410445824, 47902612878717208996830483841024

Offset: 0

Vladeta Jovovic, Mar 24 2000

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A053722, A053725, A053718, A053763.

A062250 Number of cyclic subgroups of Chevalley group A_n(2) (the group of nonsingular n X n matrices over GF(2) ).

Original entry on oeis.org

1, 5, 79, 6974, 2037136, 2890467344, 14011554132032, 325330342132674560, 27173394819858612320256, 10158190320726534408118452224, 13156630408268153048253765001412608, 80280189722884518774834501142737770774528

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 01 2001

nonn

			a(3) = 1/phi(1)+21/phi(2)+56/phi(3)+42/phi(4)+48/phi(7) = 79.

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

V. Jovovic, Cycle index of general linear group GL(n,2)

Cf. A053651 (unlabeled case), A053658, A053660, A053718, A053722, A053725, A053771-A053777, A058502, A062552, A062240.

a(n) = Sum_{d} |{g element of A_n(2): order(g)=d}|/phi(d), where phi=Euler totient function, cf. A000010.

More terms from Vladeta Jovovic, Jul 04 2001

A063386 Number of solutions of x^3=1 in general affine group AGL(n,2).

Original entry on oeis.org

1, 9, 225, 6273, 968193, 307091457, 144510377985, 338450286215169, 1535613392752345089, 11693653105154832465921, 423384155808298738368118785, 29155340360444250715547947237377

Offset: 1

Vladeta Jovovic, Jul 16 2001

nonn

V. Jovovic, Cycle index of general affine group AGL(n,2)

Cf. A063385-A063393, A062710, A053722, A053725, A053718, A053770-A053777.

A053772 Number of n X n binary matrices of order dividing 7 (i.e., number of solutions of X^7=I in GL(n,2)).

Original entry on oeis.org

1, 1, 49, 5761, 476161, 457113601, 3439085027329, 18696142934507521, 144017748317668638721, 30063679011292374997401601, 10371304522603231166854078660609, 3639433320096084212920229480292679681, 18767347744724322162378748108305552459694081

Offset: 1

Vladeta Jovovic, Mar 24 2000

nonn

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A053722, A053725, A053718.

\\ See A053725 for F(n,q,k).
F(15, 2, 7) \\ Andrew Howroyd, Jul 09 2018

a(12)-a(13) from Andrew Howroyd, Jul 09 2018

cp's OEIS Frontend

A053722 Number of n X n binary matrices of order dividing 2 (also number of solutions to X^2=I in GL(n,2)).

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A063393 Number of solutions of x^10=1 in general affine group AGL(n,2).

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A063385 Number of solutions of x^2=1 in general affine group AGL(n,2).

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A053770 Number of n X n binary matrices of order dividing 5 (i.e., number of solutions of X^5=I in GL(n,2)).

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A053777 Number of n X n binary matrices of order dividing 12 (i.e., number of solutions of X^12=I in GL(n,2)).

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A053771 Number of n X n binary matrices of order dividing 6 (i.e., number of solutions of X^6=I in GL(n,2)).

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A053773 Number of n X n binary matrices of order dividing 8 (i.e., number of solutions of X^8=I in GL(n,2)).

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A062250 Number of cyclic subgroups of Chevalley group A_n(2) (the group of nonsingular n X n matrices over GF(2) ).

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A063386 Number of solutions of x^3=1 in general affine group AGL(n,2).

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A053772 Number of n X n binary matrices of order dividing 7 (i.e., number of solutions of X^7=I in GL(n,2)).

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